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Sliding-mode pinning control of complex networks. (English) Zbl 07031757
It is considered the following model of network \[\displaystyle{\dot{x}_i=f(x_i)+\sum_{j=1,j\neq i}^Nc_{ij}a_{ij}T(x_i-x_j)+u_i(t)\;,\;i=\overline{1,N}},\] where \(x_i\) is the \(n\)-dimensional state vector of the node \(i\); \(c_{ij}>0\); \(T=diag\{\tau_1,\dots,\tau_n\}\); the connection coefficients \(a_{ij}=a_{ji}\) can be \(0\) or \(1\); \(a_{ii}=-k_i\), where \[\displaystyle{\sum_{j=1,j\neq i}^Na_{ij}=\sum_{j=1,j\neq i}^Na_{ji}=k_i.}\] The objective of the feedback control i.e. of the choice of \(u_i\) is to stabilize a homogeneous stationary state \(\bar{x}\) such that \[x_1=x_2=\ldots =x_N=\bar{x}\;,\;f(\bar{x})=0.\] The choice \[u_i=-c_{ii}d_iT(x_i-\bar{x})\;,\;d_i>0\] is called pinning strategy. This strategy is combined with sliding mode control. The stability of the achieved equilibrium is analyzed using a quadratic Lyapunov function.

MSC:
05C82 Small world graphs, complex networks (graph-theoretic aspects)
34D06 Synchronization of solutions to ordinary differential equations
93C10 Nonlinear systems in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
Software:
Matlab; Simulink
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